scholarly journals On some properties of solutions of the p-harmonic equation

Filomat ◽  
2013 ◽  
Vol 27 (4) ◽  
pp. 577-591 ◽  
Author(s):  
Sh. Chen ◽  
S. Ponnusamy ◽  
X. Wang
Keyword(s):  
Unit Disk ◽  
Connected Domain ◽  
Harmonic Mappings ◽  
Simply Connected Domain ◽  
Harmonic Equation ◽  
Simply Connected ◽  
Complex Valued ◽  
Class Of Functions

A 2p-times continuously differentiable complex-valued function ? = u + iv in a simply connected domain ? ? C is p-harmonic if ? satisfies the p-harmonic equation ?p? = 0. In this paper, we investigate the properties of p-harmonic mappings in the unit disk |z| < 1. First, we discuss the convexity, the starlikeness and the region of variability of some classes of p-harmonic mappings. Then we prove the existence of Landau constant for the class of functions of the form D? = z?z - ??z, where f is p-harmonic in |z| < 1. Also, we discuss the region of variability for certain p-harmonic mappings. At the end, as a consequence of the earlier results of the authors, we present explicit upper estimates for Bloch norm for bi- and tri-harmonic mappings.

Harmonic Mappings onto Convex Domains

1987 ◽  
Vol 39 (6) ◽  
pp. 1489-1530 ◽  
Author(s):  
Yusuf Abu-Muhanna ◽  
Glenn Schober
Keyword(s):  
Univalent Functions ◽  
Harmonic Mappings ◽  
Open Unit ◽  
Open Unit Disk ◽  
Convex Domains ◽  
Harmonic Solution ◽  
Simply Connected Domain ◽  
Image Position ◽  
Simply Connected ◽  
Complex Valued

Let D be a simply-connected domain and w0 a fixed point of D. Denote by SD the set of all complex-valued, harmonic, orientation-preserving, univalent functions f from the open unit disk U onto D with f(0) = w0. Unlike conformai mappings, harmonic mappings are not essentially determined by their image domains. So, it is natural to study the set SD.In Section 2, we give some mapping theorems. We prove the existence, when D is convex and unbounded, of a univalent, harmonic solution f of the differential equationwhere a is analytic and |a| < 1, such that f(U) ⊂ D and

Exterior Univalent Harmonic Mappings With Finite Blaschke Dilatations

1999 ◽  
Vol 51 (3) ◽  
pp. 470-487 ◽  
Author(s):  
D. Bshouty ◽  
W. Hengartner
Keyword(s):  
Partial Differential Equations ◽  
Blaschke Product ◽  
Connected Domain ◽  
Minimal Surfaces ◽  
Gauss Map ◽  
Elliptic Partial Differential Equations ◽  
Harmonic Mappings ◽  
Simply Connected Domain ◽  
Univalent Harmonic Mappings ◽  
Simply Connected

AbstractIn this article we characterize the univalent harmonic mappings from the exterior of the unit disk, Δ, onto a simply connected domain Ω containing infinity and which are solutions of the system of elliptic partial differential equations where the second dilatation function a(z) is a finite Blaschke product. At the end of this article, we apply our results to nonparametric minimal surfaces having the property that the image of its Gauss map is the upper half-sphere covered once or twice.

Quasi-isometricity and equivalent moduli of continuity of planar 1/|ω|2-harmonic mappings

Filomat ◽  
2017 ◽  
Vol 31 (2) ◽  
pp. 335-345
Author(s):  
Qi Yi ◽  
Shi Qingtian
Keyword(s):  
Quasiconformal Mapping ◽  
Connected Domain ◽  
Quasiregular Mapping ◽  
Harmonic Mappings ◽  
Moduli Of Continuity ◽  
Lipschitz Continuous ◽  
Quasihyperbolic Metric ◽  
Simply Connected Domain ◽  
Simply Connected

In this paper, we prove that 1/|?|2-harmonic quasiconformal mapping is bi-Lipschitz continuous with respect to quasihyperbolic metric on every proper domain of C\{0}. Hence, it is hyperbolic quasi-isometry in every simply connected domain on C\{0}, which generalized the result obtained in [14]. Meanwhile, the equivalent moduli of continuity for 1/|?|2-harmonic quasiregular mapping are discussed as a by-product.

scholarly journals BOUNDS FOR CERTAIN LINEAR COMBINATIONS OF THE FABER COEFFICIENTS OF FUNCTIONS ANALYTIC IN AN ELLIPSE

2007 ◽  
Vol 50 (1) ◽  
pp. 163-171
Author(s):  
E. Haliloglu
Keyword(s):  
Connected Domain ◽  
Faber Polynomials ◽  
Linear Combinations ◽  
Sharp Bounds ◽  
Simply Connected Domain ◽  
Simply Connected ◽  
Class Of Functions

AbstractLet $\varOmega$ be a bounded, simply connected domain in $\mathbb{C}$ with $0\in\varOmega$ and $\partial\varOmega$ analytic. Let $S(\varOmega)$ denote the class of functions $F(z)$ which are analytic and univalent in $\varOmega$ with $F(0)=0$ and $F'(0)=1$. Let $\{\varPhi_{n}(z)\}_{n=0}^{\infty}$ be the Faber polynomials associated with $\varOmega$. If $F(z)\in S(\varOmega)$, then $F(z)$ can be expanded in a series of the form$$ F(z)=\sum_{n=0}^{\infty}A_{n}\varPhi_{n}(z),\quad z\in\varOmega, $$in terms of the Faber polynomials. Let$$ E_{r}=\bigg\{(x,y)\in\mathbb{R}^{2}:\frac{x^{2}}{(1+(1/r^{2}))^{2}}+\frac{y^{2}}{(1-(1/r^{2}))^{2}}\lt1\bigg\}, $$where $r\gt1$.In this paper, we obtain sharp bounds for certain linear combinations of the Faber coefficients of functions $F(z)$ in $S(E_{r})$ and in certain related classes.

scholarly journals A version of Runge's theorem for the Helmholtz equation with applications to scattering theory

1989 ◽  
Vol 32 (1) ◽  
pp. 107-119 ◽  
Author(s):  
R. L. Ochs
Keyword(s):  
Wave Function ◽  
Helmholtz Equation ◽  
Connected Domain ◽  
Scattering Theory ◽  
Polar Coordinates ◽  
Differentiable Functions ◽  
Simply Connected Domain ◽  
Image Position ◽  
Simply Connected

Let D be a bounded, simply connected domain in the plane R2 that is starlike with respect to the origin and has C2, α boundary, ∂D, described by the equation in polar coordinateswhere C2, α denotes the space of twice Hölder continuously differentiable functions of index α. In this paper, it is shown that any solution of the Helmholtz equationin D can be approximated in the space by an entire Herglotz wave functionwith kernel g ∈ L2[0,2π] having support in an interval [0, η] with η chosen arbitrarily in 0 > η < 2π.

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